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Let $\{X_t\}_{t \in \mathbb R}$ be a Gaussian noise process with $X_t \sim N(0, \sigma^2)$ and $\phi$ a "sufficiently nice" function. (Say that $\phi$ is compactly supported and either smooth, continuous, or $L^2$.) I'd like to know about $$ \int_\mathbb{R} X_t \cdot \phi(t) \, dt. $$ Of course, this isn't really a well-posed question because that integral isn't defined. Can one explain this somehow as an Ito integral wrt a Brownian motion?

If one has finitely many functions $\phi_1, \dots, \phi_n$, then what can be said about the distribution of $$ \mathbf x = \left( \int_{\mathbb R} X_t \cdot \phi_1(t) \, dt, \dots, \int_{\mathbb R} X_t \cdot \phi_n(t) \, dt \right)? $$ I've read in the context of engineering papers/books that $\int_{\mathbb R} X_t \cdot \phi_1(t) \, dt$ follows the normal distribution $N(0, \|\phi_1 \|^2 \sigma^2)$, where $\| \cdot \|$ is the $L^2$ norm, however nothing is made rigorous, so I don't know why this is true. Does $\mathbf x$ follow a multivariate normal distribution with covariance matrix something like $\Sigma_{ij} = \sigma^2 \langle \phi_i, \phi_j \rangle$? (Here $\langle \cdot, \cdot \rangle$ is the usual $L^2$ inner-product.)

I apologize that this isn't a very coherent or well-formed question, but it's all that I have to work with (after searching on stochastic processes a bit). I do have experience with analysis, but not extensive experience with more formal probability; with this in mind, I'd be grateful for any references you recommend relating to my questions.

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You should think of the white noise $X_t$ as the distributional derivative of a Brownian motion $B_t$. Formally, $X_t dt = dB_t$ and the integral you are trying to compute becomes $\int \phi(t) dB_t$. But this formal expression is given a rigorous meaning as the Ito integral of $\phi$ with respect to the Brownian motion $B$.

A well-known fact is that the Ito integral of a deterministic function in $L^2$ is always Gaussian. You can see this, for example, by computing its characteristic function. The fact that the variance is given by the squared $L^2$ norm is even easier to see and follows from the Ito isometry: \begin{equation} \mathbb E \left(\int \phi(t) \; dB_t\right)^2 = \mathbb E \int \phi(t)^2 \; dt = \int \phi(t)^2 \; dt = \|\phi\|^2_{L^2}. \end{equation}

Maybe see here for a more detailed discussion.

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